The following data points are expected to follow a functional form of y = ax^b . Obtain the values of a and b by converting the functional f

Question

The following data points are expected to follow a functional form of y = ax^b . Obtain the values of a and b by converting the functional form to a linear relationship between log(x) and log(y).
x 1.21 1.35 2.40 2.75 4.50 5.10 7.1 8.1
y 1.20 1.82 5.00 8.80 19.5 32.5 55.0 80.0

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Charlie 3 weeks 2021-09-23T21:25:43+00:00 1 Answer 0

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    2021-09-23T21:26:44+00:00

    Answer:

    Step-by-step explanation:

    Given that,

    y=ax^b

    Taking the logarithm of both sides

    Log(y) =Log(ax^b)

    Applying the law of logarithm

    Log(aⁿ) =nLog(a)

    Therefore,

    Log(y) =b Log(ax)

    Also applying the product rule of logarithm

    LogAB= Log A +Log B

    Then,

    Log(y) =b{Log(a) +Log(x)}

    Log(y) =b•Log(x) +b•Log(a)

    So, this is the linear relationship

    Now,

    Using the value given

    When x=1.21, y=1.20

    Then,

    Log(y) =b{Log(a) +Log(x)}

    Log(1.2) =b{Log(a) +Log(1.21)}

    b= Log(1.2)/{Log(a) +Log(1.21)}

    Also, when x=1.35, y=1.82

    Then,

    Log(1.82) =b{Log(a) +Log(1.35)}

    Log(1.82) =b{Log(a) +Log(1.35)}

    b= Log(1.82)/{Log(a) +Log(1.35)} equation 1

    Equating the two b, since b is a constant

    Log(1.2)/{Log(a) +Log(1.21)} = Log(1.82)/{Log(a) +Log(1.35)}

    Cross multiply

    Log(1.2)•{Log(a) +Log(1.35)} = Log(1.82)•{Log(a) +Log(1.21)}

    Log(1.2)Log(a) + Log(1.2)log(1.35) = Log(1.82)Log(a) + Log(1.82)Log(1.21)

    Collect like terms

    Log(1.2)Log(a)-Log(1.82)Log(a) = Log(1.82)Log(1.21) – Log(1.2)log(1.35)

    Log(a){Log(1.2)-Log(1.82)} = Log(1.82)Log(1.21)-Log(1.2)Log(1.35)

    Log(a) = {Log(1.82)Log(1.21)-Log(1.2)Log(1.35)} / {Log(1.2)-Log(1.82)}

    Then, Log(a)=0.01121/-0.18089

    Log(a)=-0.06197

    a=antilog(-0.06197)

    a=0.867

    Then, from equation 1

    b= Log(1.82)/{Log(a) +Log(1.35)}

    b= Log(1.82)/{Log(0.867) +Log(1.35)}

    b=3.8

    Then,

    a=0.867 and

    b=3.8

    Therefore,

    y=ax^b

    y=0.867x^3.8

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